axiom of choice

statement in the language of set theory that makes it possible to form sets by choosing an element simultaneously from each member of an infinite collection of sets even when no algorithm exists for the selection. The axiom of choice has many mathematically equivalent formulations, some of which were not immediately realized to be equivalent. One version states that, given any collection of disjoint sets (sets having no common elements), there exists at least one set consisting of one element from each of the nonempty sets in the collection; collectively, these chosen elements make up the "choice set." Another common formulation is to say that for any set S there exists a function f (called a "choice function") such that, for any nonempty subset s of S, f(s) is an element of s